\magnification = 2200 %\magstep3
%\vsize=1.05\vsize

\def\UseTimesRoman{
\font\cmr=Times
\font\TR=Times at 10pt
\font\TRXII=Times at 12pt
\font\TRXIV=Times at 14pt
\font\TRXX=Times at 20pt
\font\TRXXIV=Times at 24pt
\font\TI=TimesI at 10pt     %Times Italic
\font\TB=TimesB at 10pt     %Times Bold
\font\TBI=TimesBI at 10pt   %Times Bold Italic
\font\TBIviii=TimesBI at 8pt
\font\TBIv=TimesBI at 5pt
%\font\TO=TimesO at 10pt  %Times Oblique (Times Roman, slanted 22%
  %    with EdMetrics)
\font\TO=TimesI at 10pt  %Times Oblique (Times Roman, slanted 22% with EdMetrics)

\font\TIVIII=TimesI at 8pt
\font\TRVIII=Times at 8pt
\font\TIVI=TimesI at 6pt
\font\TRVI=Times at 6pt

	     \font\tenrmscld=Times at 10 pt
        \font\sevenrmscld=Times at 7 pt
        \font\fivermscld=Times at 5 pt

        \font\teniscld=cmmi10 at 10.3 pt
        \font\seveniscld=cmmi10 at 7.21 pt
        \font\fiveiscld=cmmi10 at 5.15 pt
        \font\tensyscld=cmsy10 at 10.3 pt
        \font\sevensyscld=cmsy10 at 7.21 pt
        \font\fivesyscld=cmsy10 at 5.15 pt
        \font\tenexscld=cmex10 at 10.3 pt
        \font\tenbfscld=cmbx10 at 10.3 pt
        \font\sevenbfscld=cmbx10 at 7.21 pt
        \font\fivebfscld=cmbx10 at 5.15 pt

\font\Courier = Courier
\font\Symbol = Symbol

\def\Omega{\hbox{{\Symbol W}}}

\textfont0=\tenrmscld \scriptfont0=\sevenrmscld\scriptscriptfont0=\fivermscld
\def\rm{\fam0\tenrmscld}
\textfont1=\teniscld \scriptfont1=\seveniscld \scriptscriptfont1=\fiveiscld
\def\mit{\fam1} \def\oldstyle{\fam1\teni}
\textfont2=\tensyscld \scriptfont2=\sevensyscld \scriptscriptfont2=\fivesyscld
\def\cal{\fam2}
\textfont3=\tenexscld \scriptfont3=\tenexscld \scriptscriptfont3=\tenexscld
\def\it{\TI}
\def\sl{\TO}
\def\bf{\TB}
\def\rm{\TR}
%\def\tt{\ttCourier}
\def\tt{\Courier}
\def\abstractfont{\TRVIII}
\def\footnotefont{\TRVIII}
\def\tinyfont{\TRvi}
\def\smalltitlefont{\TRXII}
\def\titlefont{\TRXIV}
\def\bigtitlefont{\TRXX}
\def\verybigtitlefont{\TRXXIV}
\textfont9=\TBI \scriptfont9=\TBIviii \scriptscriptfont9=\TBIv
\def\mbi{\fam9}
\rm
        }

%\UseTimesRoman

\def\BR{\Bbb R}             % Besondere Buchstaben
\def\BC{\Bbb C}
\def\BI{\Bbb I}
\def\BN{\Bbb N}
\def\BQ{\Bbb Q}
\def\BS{\Bbb S}
\def\BZ{\Bbb Z}
\def\Tilde{$_{\hbox{\cmrXX \~{}}}$}
\def\ST{\hbox{\eu T }}
\def\SRS{\hbox{\eu RS}}
\def\i{\hbox{{\bf i}}}

\font\sc=cmcsc10 at 10 pt   %% or: at 10 pt
\font\eu=eusb10 %at 10 pt
\font\small=cmr8 at 8 pt
\font\cmrX=cmbx10 scaled \magstep 1 %% 12 point CM
\font\cmrXX=cmbx12 scaled \magstep 1 %%

\hsize 7 true in
\vsize 9 true in
\hoffset = -0.20 true in
\voffset -0.25 true in
\parskip=3pt

\overfullrule = 0pt



\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex

\input BoxedEPS
\SetTexturesEPSFSpecial
\HideDisplacementBoxes

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue -10pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cl {\bf About  The Dragon Curve }
\lf
\cl{ see also: Koch Snowflake, Hilbert SquareFillCurve}
\cl{ To speed up demos, press DELETE}
\Lf
The Dragon is constructed as a limit of polygonal approximations $D_n$.
These are emphasized in the 3DXM default demo and can
be described as follows:
\lf
1) $D_1$ is just a horizontal line segment.
\lf
2) $D_{n+1}$ is obtained from $D_n$ as follows:
\item{a)} Translate $D_n$, moving its end point to the origin.
\item{b)} Multiply the translated copy by $\sqrt{1/2}$.
\item{c)} Rotate the result of b) by $-45^\circ$ degrees and call
          the result $C_n$.
\item{d)} Rotate $C_n$ by $-90^\circ$ degrees and join this rotated
         copy to the end of $C_n$ to get $D_{n+1}$.

\Lf
The fact that the {\bf limit points} of a sequence of longer and longer polygons
can form a two-dimensional set is not really very surprising. What makes the
Dragon spectacular is that it is in fact a {\bf continuous curve} whose image
has positive area---properties that it shares with Hilbert's square filling curve.    
\Lf
There is a second construction of the Dragon that makes it easier to view
the limit  as a curve. Select  in the
Action Menu:\hskip0.2cm {\it Show With Previous Iteration}.
\lf
This demo shows a local construction of the Dragon: We obtain the next
iteration $D_{n+1}$ if we modify each segment of $D_{n}$ by replacing
it by an isocele $90^\circ$ triangle, alternatingly one to the left of the
segment, and the next to the right of the next segment. This description
has two advantages: 
\lf
(i) Every vertex of $D_n$ is already a point on the limit curve. Therefore one gets
a dense set of points, $c(j/2^n)$, on the limit curve $c$.
\lf
(ii) One can modify the construction by decreasing the height of the modifying
triangles from $aa = 0.5$ to $aa = 0$. The polygonal curves are, for $aa < 0.5$, polygons
without self-intersections. This makes it easier to imagine the limit as a curve. In
fact, the {\it Default Morph} shows a deformation from a segment through continuous
curves to the Dragon---more precisely, it shows the results of the $(ee = 11)$th iterations
towards those continuous limit curves.
\Lf
Finally, one can also vary the parameter $bb$ through integer values $bb = 2, 3,\dots$
to obtain other families of Fractal curves (from the Action Menu only).



\bye
 

 